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    Ufa Mathematical Journal
    Volume 3, Number 1, pp. 3-15

    Unconditional exponential bases in Hilbert spaces.


    Isaev K.P., Yulmukhametov R.S.

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    In the present paper, we consider the existence of unconditional exponential bases in general Hilbert spaces $H=H(E)$ consisting of functions defined on some set $E\subset \mathbb C$ and satisfying to conditions 1. The norm in the space $H$ is more poorly then the uniform norm on $E$, that is the following estimate holds for some constant $A$ and for any function $f$ from $H$ $$ ||f||_H\le A\sup _{z\in E}|f(z)|. $$ 2. The system of exponential functions $\{ \exp (\lambda z)$, $\lambda \in \mathbb C\}$ is the subset of $H$ and it is complete in $H$. It is proved that unconditional exponential bases cannot be constructed in $H$ if some condition is carried out. Sufficiency of the weakened condition is proved for more concrete spaces.